Derived Categories of Cubic and $V_{14}$

It is shown that, after a certain natural flop, the projectivization of the exceptional rank-$2$ vector bundle on an arbitrary smooth $V_{14}$ Fano threefold turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. Conversely, starting from a smooth cubic threefold with an instanton vector bundle of charge $2$ on it, we reconstruct a $V_{14}$ threefold. Based on the geometric properties of the above correspondence, we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth $V_{14}$ threefold and on the corresponding cubic threefold are equivalent as triangulated categories.